Question

4.) given: $overline{hv}perpoverline{gt}$, $overline{gh}congoverline{tv}$, i is mid - point of $overline{hv}$ prove: $angle gcongangle v$ statements reasons 1.) 2.) 3.) 4.) 5.) 6.)

Explanation:

Step1: Define mid - point property

Since \(I\) is the mid - point of \(\overline{HV}\), we have \(HI = IV\).

Step2: Define perpendicular property

Given \(\overline{HV}\perp\overline{GT}\), \(\angle GIH=\angle TIV = 90^{\circ}\) (Definition of perpendicular lines).

Step3: Identify given side equality

We are given that \(GH = TV\).

Step4: Apply Hypotenuse - Leg (HL) theorem

In right - triangles \(\triangle GIH\) and \(\triangle TIV\), we have \(GH = TV\) (given) and \(HI = IV\) (mid - point property), so \(\triangle GIH\cong\triangle TIV\) (HL theorem for right - triangles).

Step5: Use congruent triangle property

If \(\triangle GIH\cong\triangle TIV\), then corresponding angles are congruent.

Step6: Conclude angle congruence

\(\angle G\cong\angle V\) (Corresponding parts of congruent triangles are congruent).

Answer:

Statements	Reasons
2. \(\overline{HV}\perp\overline{GT}\)	Given
3. \(GH = TV\)	Given
4. \(HI = IV\)	Definition of mid - point
5. \(\angle GIH=\angle TIV = 90^{\circ}\)	Definition of perpendicular lines
6. \(\triangle GIH\cong\triangle TIV\)	Hypotenuse - Leg (HL) theorem
7. \(\angle G\cong\angle V\)	Corresponding parts of congruent triangles are congruent